All of the 3rd grade teachers and students from Almond went on a field trip to an art museum. Tickets were $$8.50$ each for teachers and $$2.50$ each for students, and the group paid $$39.50$ in total. The next month, the same group visited a science museum where the tickets cost $$34.00$ each for teachers and $$13.00$ each for students, and the group paid $$185.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+2.5y = 39.5}$ ${34x+13y = 185}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-34x-10y = -158}$ ${34x+13y = 185}$ Add the top and bottom equations together. $ 3y = 27 $ $ y = \dfrac{27}{3}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {8.5x+2.5y = 39.5}$ to find $x$ ${8.5x + 2.5}{(9)}{= 39.5}$ $8.5x+22.5 = 39.5$ $8.5x = 17$ $x = \dfrac{17}{8.5}$ ${x = 2}$ You can also plug ${y = 9}$ into $ {34x+13y = 185}$ and get the same answer for $x$ ${34x + 13}{(9)}{= 185}$ ${x = 2}$ There were $2$ teachers and $9$ students on the field trips.